Limiting curves for the dyadic odometer

A. R. Minabutdinov

Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XXX, Tome 481 (2019), pp. 74-86

Résumé

A limiting curve of a stationary process in discrete time was defined by É. Janvresse, T. de la Rue, and Y. Velenik as the uniform limit of the functions $$ t\mapsto \big(S(tl_n) - tS(l_n)\big)/R_n \in C([0, 1]), $$ where $S$ stands for the piecewise linear extension of the partial sum, $R_n := \sup |S(tl_n) - tS(l_n))|$, and $(l_n) = (l_n(\omega))$ is a suitable sequence of integers. We determine the limiting curves for the stationary sequence $(f\circ T^n(\omega))$ where $T$ is the dyadic odometer on $\{0,1\}^{\mathbb{N}}$ and $$f((\omega_i)) = \sum\limits_{i\geq 0}\omega_iq^{i+1}$$ for $1/2 < |q| < 1.$ Namely, we prove that for a.e. $\omega$ there exists a sequence $(l_n(\omega))$ such that the limiting curve exists and is equal to $(-1)$ times the Tagaki–Landsberg function with parameter $1/2q.$ The result can be obtained as a corollary of a generalization of the Trollope–Delange formula to the $q$-weighted case.

Bibliographie

[1] A. M. Vershik, “Ravnomernaya algebraicheskaya approksimatsiya operatorov sdviga i umnozheniya”, DAN SSSR, 259:3 (1981), 526–529 | Zbl

[2] A. M. Vershik, “Teorema o markovskoi periodicheskoi approksimatsii v ergodicheskoi teorii”, Zap. nauchn. semin. LOMI, 115, 1982, 72–82 | Zbl

[3] A. M. Vershik, B. Solomyak, “Adicheskaya realizatsiya preobrazovaniya Morsa i prodolzhenie ego deistviya na solenoid”, Zap. nauchn. semin. POMI, 360, 2008, 70–90 | MR

[4] A. P. Minabutdinov, I. E. Manaev, “Funktsiya Kruskala-Katony, posledovatelnost Konveya, krivaya Takagi i avtomorfizm Paskalya”, Zap. nauchn. semin. POMI, 411, 2013, 135–147

[5] A. A. Lodkin, A. P. Minabutdinov, “Predelnye krivye dlya avtomorfizma Paskalya”, Zap. nauchn. semin. POMI, 437, 2015, 145–183

[6] A. P. Minabutdinov, “Teorema suschestvovaniya predelnykh krivykh dlya polinomialnykh adicheskikh avtomorfizmov”, Zap. nauchn. semin. POMI, 448 (2016), 177–200

[7] S. Bailey, Dynamical properties of some non-stationary, non-simple Bratteli–Vershik systems, Ph.D. thesis, University of North Carolina, Chapel Hill, 2006 | MR

[8] A. Minabutdinov, Limiting curves for the dyadic odometer and the generalized Trollope–Delange formula, 2018, arXiv: 1801.03120 | MR

[9] P. Allaart, K. Kawamura, “The Takagi function: a survey”, Real Anal. Exchange, 37:1 (2012), 1–54 | DOI | MR | Zbl

[10] M. Barnsley, “Fractal functions and interpolation”, Constr. Approx., 2 (1986), 303–329 | DOI | MR | Zbl

[11] H. Delange, “Sur la fonction sommatoire de la fonction “somme des chiffres””, Enseign. Math., 21:2 (1975), 31–47 | MR | Zbl

[12] R. Girgensohn, “Nowhere differentiable solutions of a system of functional equations”, Aequationes Math., 47:1 (1994), 89–99 | DOI | MR | Zbl

[13] R. Girgensohn, “Digital Sums and functional equations”, Integers, 12:1 (2012), 141–160 | DOI | MR | Zbl

[14] É. Janvresse, T. de la Rue, “The Pascal adic transformation is loosely Bernoulli”, Ann. Inst. H. Poincaré Probab. Statist, 40:2 (2004), 133–139 | DOI | MR | Zbl

[15] É. Janvresse, T. de la Rue, Y. Velenik, “Self-similar corrections to the ergodic theorem for the Pascal-adic transformation”, Stoch. Dyn., 5:1 (2005), 1–25 | DOI | MR | Zbl

[16] S. Kakutani, “A problem of equidistribution on the unit interval [0, 1]”, Lect. Notes Math., 541, 1976, 369–375 | DOI | MR | Zbl

[17] J. C. Lagarias, “The Takagi function and its properties”, Functions in Number Theory and Their Probabilistic Aspects, 2012, 153–189 | MR | Zbl

[18] G. Landsberg, “Über die Differentiierbarkeit stetiger Funktionen”, Jahresber. Deutschen Math. Verein., 17 (1908), 46–51 | Zbl

[19] A. Lodkin, A. Minabutdinov, “Limiting curves for the Pascal adic transformation”, J. Math. Sci., 216:1 (2016), 94–119 | DOI | MR | Zbl

[20] X. Méla, K. Petersen, “Dynamical properties of the Pascal adic transformation”, Ergodic Theory Dynam. Systems, 25:1 (2005), 227–256 | DOI | MR | Zbl

[21] A. Minabutdinov, “Limiting curves for polynomial adic systems”, Zap. Nauchn. Semin. POMI, 448, 2016, 177–200

[22] A. Nogueira, E. H. El Abdalaoui, M. El Machkouri, “A criterion of weak mixing property”, École de Théorie Ergodique, Sémin. Congr., 20, Soc. Math. France, Paris, 2010, 105–111 | MR | Zbl

[23] G. de Rham, “Sur quelques courbes définies par des équations fonctionnelles”, Rend. Sem. Mat., 16 (1956), 101–113 | MR

[24] T. Takagi, “A simple example of the continuous function without derivative”, Proc. Phys.-Math. Soc., 1 (1903), 176–177 | Zbl

[25] E. Trollope, “An explicit expression for binary digital sums”, Math. Mag., 41 (1968), 21–25 | DOI | MR | Zbl

[26] A. M. Vershik, “Remarks on Pascal automorphism and infinite ergodic theory”, Armenian J. Math., 7:2 (2015), 85–96 | MR | Zbl